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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 715026, 8 pages
Research Article

Optimal Control for a Family of Systems in Novel State Derivative Space Form with Experiment in a Double Inverted Pendulum System

Department of Electrical Engineering, I-Shou University, Kaohsiung 84001, Taiwan

Received 13 June 2013; Accepted 29 July 2013

Academic Editor: Chang-Hua Lien

Copyright © 2013 Yuan-Wei Tseng and Jer-Guang Hsieh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Optimal control for a family of systems in novel state derivative space form, abbreviated as SDS systems in this study, is proposed. The first step in deriving optimal control laws for SDS systems is to form an augmented cost functional. It turns out that novel differential Lagrange multipliers must be used to adjoin SDS system constraints (namely, the dynamical equations of the control system) to the integrand of the original cost functional which is a function of state derivatives. This not only eases our derivation but also makes our derivation parallel to that for systems in standard state space form. We will show via a real electric circuit that optimal control for a class of descriptor systems with impulse modes can easily be carried out using our design method. It will be shown that linear quadratic regulator (LQR) design for linear time-invariant SDS systems using state derivative feedback can be obtained via an algebraic Riccati equation. Furthermore, this optimal state derivative feedback may also be implemented using an equivalent state feedback. This is useful in real situations when only states but not the state derivatives are available for measurement. The LQR design for a double inverted pendulum system is implemented to illustrate the use of our method.